This can be the main finished survey of the mathematical lifetime of the mythical Paul Erd? s, probably the most flexible and prolific mathematicians of our time. For the 1st time, the entire major components of Erd? s' examine are lined in one venture. as a result of overwhelming reaction from the mathematical neighborhood, the venture now occupies over 900 pages, prepared into volumes.

But, since A ∩ B ⊆ A for all A and B, it is sufficient to prove that A ⊆ A ∩ B. So, if x ∈ A, it follows from (i) that x ∈ B and therefore x ∈ A ∩ B. Hence, A ⊆ A ∩ B. 38 SET THEORY To prove that (ii) implies (iii), let’s assume that A ∩ B = A holds. Then, A ∪ B = (A ∩ B) ∪ B = (A ∪ B) ∩ (B ∪ B) = (A ∪ B) ∩ B = B Finally, to prove that (iii) implies (i), we assume that A ∪ B = B holds. Then, since A ⊆ A ∪ B for all A and B, it follows that A ⊆ B. 29 Determine (A ∩ B) ∪ (Ac ∩ Cc ) Solution If A = [0, 1), then Ac = (−∞, 0) ∪ [1, ∞).

19 A set of all natural numbers N is infinite. 20 A set of all integers Z is infinite. ◾ We will discuss the intricacies of infinite sets in a little while. 10 We say that two sets A and B are equivalent (or equinumerous) or that they have the same cardinality, and we write A∼B iff |A| = |B| Following Cantor, we say that cardinal number of a set A is what A has in common with all sets equivalent to A. 21 Given sets A = {1, 2, 3}, B = {a, b, c}, and C = {b, c, a}, we say that A ∼ B, and A ∼ C, but only B = C.